scholarly journals The critical point of the transition to turbulence in pipe flow

2018 ◽  
Vol 839 ◽  
pp. 76-94 ◽  
Author(s):  
Vasudevan Mukund ◽  
Björn Hof
Keyword(s):  
Steady State ◽  
Critical Point ◽  
Pipe Flow ◽  
Initial Conditions ◽  
Flow Patterns ◽  
Finite Amplitude ◽  
Decay Rates ◽  
Transition To Turbulence ◽  
Intermittent Flow ◽  
Spatio Temporal

In pipes, turbulence sets in despite the linear stability of the laminar Hagen–Poiseuille flow. The Reynolds number ($Re$) for which turbulence first appears in a given experiment – the ‘natural transition point’ – depends on imperfections of the set-up, or, more precisely, on the magnitude of finite amplitude perturbations. At onset, turbulence typically only occupies a certain fraction of the flow, and this fraction equally is found to differ from experiment to experiment. Despite these findings, Reynolds proposed that after sufficiently long times, flows may settle to steady conditions: below a critical velocity, flows should (regardless of initial conditions) always return to laminar, while above this velocity, eddying motion should persist. As will be shown, even in pipes several thousand diameters long, the spatio-temporal intermittent flow patterns observed at the end of the pipe strongly depend on the initial conditions, and there is no indication that different flow patterns would eventually settle to a (statistical) steady state. Exploiting the fact that turbulent puffs do not age (i.e. they are memoryless), we continuously recreate the puff sequence exiting the pipe at the pipe entrance, and in doing so introduce periodic boundary conditions for the puff pattern. This procedure allows us to study the evolution of the flow patterns for arbitrary long times, and we find that after times in excess of $10^{7}$ advective time units, indeed a statistical steady state is reached. Although the resulting flows remain spatio-temporally intermittent, puff splitting and decay rates eventually reach a balance, so that the turbulent fraction fluctuates around a well-defined level which only depends on $Re$. In accordance with Reynolds’ proposition, we find that at lower $Re$ (here 2020), flows eventually always resume to laminar, while for higher $Re$ (${\geqslant}2060$), turbulence persists. The critical point for pipe flow hence falls in the interval of $2020<Re<2060$, which is in very good agreement with the recently proposed value of $Re_{c}=2040$. The latter estimate was based on single-puff statistics and entirely neglected puff interactions. Unlike in typical contact processes where such interactions strongly affect the percolation threshold, in pipe flow, the critical point is only marginally influenced. Interactions, on the other hand, are responsible for the approach to the statistical steady state. As shown, they strongly affect the resulting flow patterns, where they cause ‘puff clustering’, and these regions of large puff densities are observed to travel across the puff pattern in a wave-like fashion.

scholarly journals Transition to turbulence in pulsating pipe flow

2017 ◽  
Vol 831 ◽  
pp. 418-432 ◽  
Author(s):  
Duo Xu ◽  
Sascha Warnecke ◽  
Baofang Song ◽  
Xingyu Ma ◽  
Björn Hof
Keyword(s):  
Pipe Flow ◽  
Oscillation Amplitude ◽  
Intermediate Frequency ◽  
Finite Amplitude ◽  
Decay Rates ◽  
Transition To Turbulence ◽  
Pulsation Frequency ◽  
Onset Of Turbulence ◽  
Frequency Regime ◽  
The Impact

Fluid flows in nature and applications are frequently subject to periodic velocity modulations. Surprisingly, even for the generic case of flow through a straight pipe, there is little consensus regarding the influence of pulsation on the transition threshold to turbulence: while most studies predict a monotonically increasing threshold with pulsation frequency (i.e. Womersley number, $\unicode[STIX]{x1D6FC}$), others observe a decreasing threshold for identical parameters and only observe an increasing threshold at low $\unicode[STIX]{x1D6FC}$. In the present study we apply recent advances in the understanding of transition in steady shear flows to pulsating pipe flow. For moderate pulsation amplitudes we find that the first instability encountered is subcritical (i.e. requiring finite amplitude disturbances) and gives rise to localized patches of turbulence (‘puffs’) analogous to steady pipe flow. By monitoring the impact of pulsation on the lifetime of turbulence we map the onset of turbulence in parameter space. Transition in pulsatile flow can be separated into three regimes. At small Womersley numbers the dynamics is dominated by the decay turbulence suffers during the slower part of the cycle and hence transition is delayed significantly. As shown in this regime thresholds closely agree with estimates based on a quasi-steady flow assumption only taking puff decay rates into account. The transition point predicted in the zero $\unicode[STIX]{x1D6FC}$ limit equals to the critical point for steady pipe flow offset by the oscillation Reynolds number (i.e. the dimensionless oscillation amplitude). In the high frequency limit on the other hand, puff lifetimes are identical to those in steady pipe flow and hence the transition threshold appears to be unaffected by flow pulsation. In the intermediate frequency regime the transition threshold sharply drops (with increasing $\unicode[STIX]{x1D6FC}$) from the decay dominated (quasi-steady) threshold to the steady pipe flow level.

The effect of rigid rotation on the finite-amplitude stability of pipe flow at high Reynolds number

1984 ◽  
Vol 148 ◽  
pp. 193-205 ◽  
Author(s):  
T. R. Akylas ◽  
J.-P. Demurger
Keyword(s):  
Reynolds Number ◽  
Pipe Flow ◽  
High Reynolds Number ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Rigid Rotation ◽  
Transition To Turbulence ◽  
Neutral Stability ◽  
Axisymmetric Mode ◽  
High Reynolds

A theoretical study is made of the stability of pipe flow with superimposed rigid rotation to finite-amplitude disturbances at high Reynolds number. The non-axisymmetric mode that requires the least amount of rotation for linear instability is considered. An amplitude expansion is developed close to the corresponding neutral stability curve; the appropriate Landau constant is calculated. It is demonstrated that the flow exhibits nonlinear subcritical instability, the nonlinear effects being particularly strong owing to the large magnitude of the Landau constant. These findings support the view that a small amount of extraneous rotation could play a significant role in the transition to turbulence of pipe flow.

scholarly journals Transition to turbulence when the Tollmien–Schlichting and bypass routes coexist

2019 ◽  
Vol 880 ◽  
Author(s):  
Stefan Zammert ◽  
Bruno Eckhardt
Keyword(s):  
Initial Conditions ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Transition To Turbulence ◽  
Stable Region ◽  
Bypass Transition ◽  
Parallel Plates ◽  
Tollmien Schlichting

Plane Poiseuille flow, the pressure-driven flow between parallel plates, shows a route to turbulence connected with a linear instability to Tollmien–Schlichting (TS) waves, and another route, the bypass transition, that can be triggered with finite-amplitude perturbation. We use direct numerical simulations to explore the arrangement of the different routes to turbulence among the set of initial conditions. For plates that are a distance $2H$ apart, and in a domain of width $2\unicode[STIX]{x03C0}H$ and length $2\unicode[STIX]{x03C0}H$, the subcritical instability to TS waves sets in at $Re_{c}=5815$ and extends down to $Re_{TS}\approx 4884$. The bypass route becomes available above $Re_{E}=459$ with the appearance of three-dimensional, finite-amplitude travelling waves. Below $Re_{c}$, TS transition appears for a tiny region of initial conditions that grows with increasing Reynolds number. Above $Re_{c}$, the previously stable region becomes unstable via TS waves, but a sharp transition to the bypass route can still be identified. Both routes lead to the same turbulent state in the final stage of the transition, but on different time scales. Similar phenomena can be expected in other flows where two or more routes to turbulence compete.

scholarly journals Localised turbulence in a circular pipe flow with gradual expansion

2015 ◽  
Vol 771 ◽  
Author(s):  
Kamal Selvam ◽  
Jorge Peixinho ◽  
Ashley P. Willis
Keyword(s):  
Reynolds Number ◽  
Pipe Flow ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Finite Amplitude ◽  
Transition To Turbulence ◽  
Circular Pipe ◽  
Recirculation Region ◽  
Circular Pipe Flow ◽  
Gradual Expansion

We report the results of three-dimensional direct numerical simulations for incompressible viscous fluid in a circular pipe flow with a gradual expansion. At the inlet, a parabolic velocity profile is applied together with a constant finite-amplitude perturbation to represent experimental imperfections. Initially, at low Reynolds number, the solution is steady. As the Reynolds number is increased, the length of the recirculation region near the wall grows linearly. Then, at a critical Reynolds number, a symmetry-breaking bifurcation occurs, where linear growth of asymmetry is observed. Near the point of transition to turbulence, the flow experiences oscillations due to a shear layer instability for a narrow range of Reynolds numbers. At higher Reynolds numbers, the recirculation region breaks into a turbulent state which remains spatially localised and unchanged when the perturbation is removed from the flow. Spatial correlation analysis suggests that the localised turbulence in the gradual expansion possesses a different flow structure from the turbulent puff of uniform pipe flow.

An investigation of transition to turbulence in bounded oscillatory Stokes flows Part 2. Numerical simulations

1991 ◽  
Vol 225 ◽  
pp. 423-444 ◽  
Author(s):  
R. Akhavan ◽  
R. D. Kamm ◽  
A. H. Shapiro
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Transition To Turbulence ◽  
Quasi Equilibrium ◽  
Two Dimensional ◽  
Spectral Techniques ◽  
Periodic Steady State

The stability of oscillatory channel flow to different classes of infinitesimal and finite-amplitude two- and three-dimensional disturbances has been investigated by direct numerical simulations of the Navier–Stokes equations using spectral techniques. All infinitesimal disturbances were found to decay monotonically to a periodic steady state, in agreement with earlier Floquet theory calculations. However, before reaching this periodic steady state an infinitesimal disturbance introduced in the boundary layer was seen to experience transient growth in accordance with the predictions of quasi-steady theories for the least stable eigenmodes of the Orr–Sommerfeld equation for instantaneous ‘frozen’ profiles. The reason why this growth is not sustained in the periodic steady state is explained. Two-dimensional infinitesimal disturbances reaching finite amplitudes were found to saturate in an ordered state of two-dimensional quasi-equilibrium waves that decayed on viscous timescales. No finite-amplitude equilibrium waves were found in our cursory study. The secondary instability of these two-dimensional finite-amplitude quasi-equilibrium states to infinitesimal three-dimensional perturbations predicts transitional Reynolds numbers and turbulent flow structures in agreement with experiments.

Dynamical systems and the transition to turbulence in linearly stable shear flows

2007 ◽  
Vol 366 (1868) ◽  
pp. 1297-1315 ◽  
Author(s):  
Bruno Eckhardt ◽  
Holger Faisst ◽  
Armin Schmiegel ◽  
Tobias M Schneider
Keyword(s):  
Couette Flow ◽  
Coherent Structures ◽  
Pipe Flow ◽  
Temporal Evolution ◽  
Initial Conditions ◽  
Transition To Turbulence ◽  
Plane Couette Flow ◽  
Numerical Studies ◽  
Lifetime Studies ◽  
Transition Scenario

Plane Couette flow and pressure-driven pipe flow are two examples of flows where turbulence sets in while the laminar profile is still linearly stable. Experiments and numerical studies have shown that the transition has features compatible with the formation of a strange saddle rather than an attractor. In particular, the transition depends sensitively on initial conditions and the turbulent state is not persistent but has an exponential distribution of lifetimes. Embedded within the turbulent dynamics are coherent structures, which transiently show up in the temporal evolution of the turbulent flow. Here we summarize the evidence for this transition scenario in these two flows, with an emphasis on lifetime studies in the case of plane Couette flow and on the coherent structures in pipe flow.

Finite-amplitude thresholds for transition in pipe flow

2007 ◽  
Vol 582 ◽  
pp. 169-178 ◽  
Author(s):  
J. PEIXINHO ◽  
T. MULLIN
Keyword(s):  
Experimental Study ◽  
Pipe Flow ◽  
Pipe Wall ◽  
Critical Value ◽  
Finite Amplitude ◽  
The Other ◽  
Transition To Turbulence ◽  
Constant Mass ◽  
Hairpin Vortices ◽  
Order Of Magnitude

We report the results of an experimental study of the finite-amplitude thresholds for transition to turbulence in a constant mass flux pipe flow. The flow was perturbed using small impulsive jets and push–pull disturbances from holes in the pipe wall. The flux of the disturbance is used to define an amplitude for the perturbation and the critical value required to cause transition scales in proportion to Re−1 for jets. In this case, the transition is catastrophic and the scaling suggests a simple balance between inertia and viscosity. On the other hand, the threshold scales as Re−1.3 or Re−1.5 for push–pull disturbances with the precise value depending on the orientation of the perturbation. Further, the amplitudes required to cause transition are typically an order of magnitude smaller than for jets. When the push–pull perturbation was applied in the oblique direction, streaks and hairpin vortices appeared during the growth phase of the disturbance. The scaling of the threshold and the growth of structures are both consistent with ideas associated with temporary algebraic growth.

An Exploratory Experimental Technique to Predict Two-Phase Flow Pattern From Vibration Response

2013 ◽  
Author(s):  
Luis Enrique Ortiz-Vidal ◽  
Oscar M. H. Rodriguez ◽  
Njuki Mureithi
Keyword(s):  
Flow Pattern ◽  
Experimental Technique ◽  
Pipe Flow ◽  
Two Phase Flow ◽  
Flow Patterns ◽  
Vibration Response ◽  
Structural Vibration ◽  
Intermittent Flow ◽  
Phase Flow ◽  
Two Phase

Gas-liquid pipe flow is common in nuclear, gas & oil, refrigeration and power generation industries, where gas-liquid mixtures are transported in piping systems. The mixtures flows in different flow patterns, such as bubbly, slug and annular, generating dynamic fluid forces which may induce structural vibration. In many industrial cases, Flow-Induced Vibrations (FIV) are an intrinsic part of the piping operation and does not present risks that may lead to structural component failures. In this sense, the information available on this topic is quite scanty. In this paper, we present an in-depth discussion about the phenomenology of the FIV due to two-phase pipe flow. A set of 32 two-phase horizontal flow conditions was collected, including bubbly, slug and dispersed flow-patterns. The homogeneous mixture velocity J was in the range of 0.5 to 25 m/s, with homogeneous void fractions of β = 10%, 25%, 50%, 75% and 95%. Signals of acceleration were acquired to correlate pipe vibration and two-phase flow parameters. Results show higher acceleration levels in slug and dispersed than in bubbly flow. We find that the acceleration frequency response contains useful information of the flow. Comparisons with single-phase flow are also presented. Finally, an exploratory experimental technique to predict two-phase flow pattern from vibration response based on the combination resonance caused by both single and two-phase flow is proposed. The results indicate that the proposed-technique is acceptable to recognize intermittent flow patterns in two-phase flow.

scholarly journals Experimental and theoretical progress in pipe flow transition

2008 ◽  
Vol 366 (1876) ◽  
pp. 2671-2684 ◽  
Author(s):  
A.P Willis ◽  
J Peixinho ◽  
R.R Kerswell ◽  
T Mullin
Keyword(s):  
Pipe Flow ◽  
Quantitative Agreement ◽  
Travelling Wave ◽  
Turbulent Pipe Flow ◽  
Transition To Turbulence ◽  
Reynolds Number Flow ◽  
Threshold Amplitude ◽  
Number Flow ◽  
The Stability ◽  
Laminar State

There have been many investigations of the stability of Hagen–Poiseuille flow in the 125 years since Osborne Reynolds' famous experiments on the transition to turbulence in a pipe, and yet the pipe problem remains the focus of attention of much research. Here, we discuss recent results from experimental and numerical investigations obtained in this new century. Progress has been made on three fundamental issues: the threshold amplitude of disturbances required to trigger a transition to turbulence from the laminar state; the threshold Reynolds number flow below which a disturbance decays from turbulence to the laminar state, with quantitative agreement between experimental and numerical results; and understanding the relevance of recently discovered families of unstable travelling wave solutions to transitional and turbulent pipe flow.

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